The existence of a unique solution to a mixed control problem for Sobolev-type equations

This article studies a mixed control problem for Sobolev-type equations in the case of a relatively radial operator. We use the Showalter–Sidorov initial condition. The difference in the statement of our problem from those studied previously by other researchers amounts to the form of the quality functional, which, in the authors' opinion, is more adequate to model applications in economics and technology. We prove an existence and uniqueness theorem for the solution to this problem.